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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2zrng0 | Structured version Visualization version GIF version |
Description: The additive identity of R is the complex number 0. (Contributed by AV, 11-Feb-2020.) |
Ref | Expression |
---|---|
2zrng.e | ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} |
2zrngbas.r | ⊢ 𝑅 = (ℂfld ↾s 𝐸) |
Ref | Expression |
---|---|
2zrng0 | ⊢ 0 = (0g‘𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncrng 20112 | . . 3 ⊢ ℂfld ∈ CRing | |
2 | crngring 19302 | . . 3 ⊢ (ℂfld ∈ CRing → ℂfld ∈ Ring) | |
3 | ringmnd 19300 | . . 3 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
4 | 1, 2, 3 | mp2b 10 | . 2 ⊢ ℂfld ∈ Mnd |
5 | 2zrng.e | . . 3 ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} | |
6 | 5 | 0even 44555 | . 2 ⊢ 0 ∈ 𝐸 |
7 | ssrab2 4007 | . . . 4 ⊢ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} ⊆ ℤ | |
8 | 5, 7 | eqsstri 3949 | . . 3 ⊢ 𝐸 ⊆ ℤ |
9 | zsscn 11977 | . . 3 ⊢ ℤ ⊆ ℂ | |
10 | 8, 9 | sstri 3924 | . 2 ⊢ 𝐸 ⊆ ℂ |
11 | 2zrngbas.r | . . 3 ⊢ 𝑅 = (ℂfld ↾s 𝐸) | |
12 | cnfldbas 20095 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
13 | cnfld0 20115 | . . 3 ⊢ 0 = (0g‘ℂfld) | |
14 | 11, 12, 13 | ress0g 17931 | . 2 ⊢ ((ℂfld ∈ Mnd ∧ 0 ∈ 𝐸 ∧ 𝐸 ⊆ ℂ) → 0 = (0g‘𝑅)) |
15 | 4, 6, 10, 14 | mp3an 1458 | 1 ⊢ 0 = (0g‘𝑅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 ∃wrex 3107 {crab 3110 ⊆ wss 3881 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 · cmul 10531 2c2 11680 ℤcz 11969 ↾s cress 16476 0gc0g 16705 Mndcmnd 17903 Ringcrg 19290 CRingccrg 19291 ℂfldccnfld 20091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-cmn 18900 df-mgp 19233 df-ring 19292 df-cring 19293 df-cnfld 20092 |
This theorem is referenced by: 2zrngagrp 44567 |
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